Polynomials with Multiple Zeros and Solvable Dynamical Systems Including Models in the Plane with Polynomial Interactions

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N N N N−m pN (z; t)= z + ym (t) z = [z − xn (t)] , (1) m=1 n=1 X Y to the time-evolution of its N coefficients ym (t). Indeed, if the time evolution of the N coefficients ym (t) is determined by a system of Ordinary Differential Equations (ODEs) which is itself solvable, then the corresponding time-evolution of the N zeros xn (t) is also solvable, via the following 3 steps: (i) given the initial values xn (0) , the corresponding initial values ym (0) can be obtained from the explicit formulas expressing the coefficients ym of a polynomial in terms of its zeros xn reading (for all time, hence in particular at t = 0)

N M m ym (t) = (−1) [xnℓ (t)] , m =1, 2, ..., N ; (2) ≤ 1 2 m≤ ( ) 1 n

2 the property to be isochronous or asymptotically isochronous (see for instance [2] [3] [4]). The viability of this technique to identify solvable dynamical systems de- pends of course on the availability of an explicit method to relate the time- evolution of the N zeros of a polynomial to the corresponding time-evolution of its N coefficients. Such a method was indeed provided in [1], opening the way to the identification of a vast class of algebraically solvable dynamical systems (see also [5] and references therein); but that approach was essentially restricted to the consideration of linear time evolutions of the coefficients ym (t). A development allowing to lift this quite strong restriction emerged relatively recently [6], by noticing the validity of the identity

−1 N N x˙ = − (x − x ) y˙ (x )N−m (3) n  n ℓ  m n 6 m=1 ℓ=1Y, ℓ=n X h i   which provides a convenient explicit relationship among the time evolutions of the N zeros xn (t) and the N coefficients ym (t) of the generic polynomial (1). This allowed a major enlargement of the class of algebraically solvable dynamical systems identifiable via this approach: for many examples see [7] and references therein. Remark 1-2. Analogous identities to (3) have been identified for higher time-derivatives [8] [9] [7]; but in this paper we restrict our treatment to dynamical systems characterized by first-order ODEs, postponing the treatment of dynamical systems characterized by higher-order ODEs (see Section 6). A new twist of this approach was then provided by its extension to nongeneric polynomials featuring—for all time—multiple zeros. The first step in this direction focussed on time-dependent polynomials featuring for all time a single double zero [10]; and subsequently significant progress has been made to treat the case of polynomials featuring a single zero of arbitrary multiplicity [11]. In Section 2 of the present paper a convenient method is provided which is suitable to treat the most general case of polynomials featuring an arbitrary number of zeros each of which features an arbitrary multiplicity. While all these developments might appear to mimic scholastic exercises analogous to the discussion among medieval scholars of how many angels might dance simul- taneously on the point of a needle, they do indeed provide new tools to identify new dynamical systems featuring interesting time evolutions (including systems displaying remarkable behaviors such as isochrony or asymptotic isochrony: see for instance [10] [11]); dynamical systems which—besides their intrinsic mathe- matical interest—are quite likely to play significant roles in applicative contexts. Such developments shall be reported in future publications. In the present paper we focus on another twist of this approach to identify new solvable dynamical systems which was introduced quite recently [12]. It is again based on the relations among the time-evolution of the coefficients and the zeros of time- dependent polynomials [6] [7] with multiple roots (see [10], [11] and above); but (as in [12]) by restricting attention to such polynomials featuring only 2 zeros.

3 Again, this might seem such a strong limitation to justify the doubt that the results thereby obtained be of much interest. But the eﬀect of this restriction is to open the possibility to identify algebraically solvable dynamical models characterized by the following system of 2 ODEs,

(n) x˙ n = P (x1, x2) , n =1, 2 , (4) (n) with P (x1, x2) two polynomials in the two dependent variables x1 (t) and x2 (t); hence systems of considerable interest, both from a theoretical and an applicative point of view (see [12] and references quoted there). This development is detailed in the following Section 3 by treating a speciﬁc example. In Section 4 we report—without detailing their derivation, which is rather obvious on the basis of the treatment provided in Section 3—many other such solvable models (see (4); but in some cases the right-hand side of these equations are not quite polynomial); and a simple technique allowing additional extensions of these models—making them potentially more useful in applicative contexts—is outlined in Section 5, by detailing its applicability in a particularly interesting case. Hence researchers primarily interested in applications of such systems of ODEs might wish to take ﬁrst of all a quick look at these 2 sections. Finally, Section 6 outlines future developments of this research line; and some material useful for the treatment provided in the body of this paper is reported in 2 Appendices.

2 Properties of nongeneric time-dependent polynomials featuring N zeros, each of arbitrary multiplicity

In this Section 2 we focus on time-dependent (monic) polynomials featuring for all time N diﬀerent zeros xn (t), each of which with the arbitrarily assigned (of course time-independent) multiplicity µn. They are of course deﬁned as follows:

M N M M−m µn PM (z; t)= z + ym (t) z = {[z − xn (t)] } . (5a) m=1 n=1 X Y Here the N positive integers µn are a priori arbitrary. It is obvious that this formula implies that the degree of this polynomial PM (z; t) is

M = (µn) . (5b) n=1 X It is plain that there exist explicit formulas—generalizing (2)—expressing the M coeﬃcients ym (t) in terms of the N zeros xn (t); for instance clearly

N N M µn y1 (t)= − [µnxn (t)] , yM (t) = (−1) {[xn (t)] } , (6) n=1 n=1 X Y

4 and see other examples below. It is also plain that, while N zeros xn and their multiplicities µn can be arbitrarily assigned in order to define the polynomial (5), this is not the case for the M coefficients ym: generally—for any given assignment of the N multiplicities µn—only N of them can be arbitrarily assigned, thereby determining (via algebraic operations) the N zeros xn and the remaining M −N other coefficients ym. Remark 2-1. The generic polynomial (1), of degree N and featuring N different zeros xn and N coefficients ym, generally implies that the set of its N coefficients ym is an N-vector ~y = (y1, ..., yN ) , while the set of its N zeros xn is instead an unordered set of N numbers xn. This however is not quite true in the case of a time-dependent generic polynomial which features—as those generally considered in this paper—a continuous time-dependence of its coefficients and zeros; then the set of its N zeros xn (0) at the initial time t = 0 should be generally considered an unordered set, but for all subsequent time, t > 0, the set of its N zeros xn (t) is an ordered set, the assignment of the index n to xn (t) being no more arbitrary but rather determined by continuity in t (at least provided during the time evolution no collision of two or more different zeros occur, in which case the identities of these zeros get to some extent lost because their identities may be exchanged, becoming undetermined). The situation is quite different in the case of a nongeneric polynomial such as (5): then zeros having different multiplicities are intrinsically different, for instance if all the multiplicities µn are different among themselves, µn 6= µℓ if n 6= ℓ, then clearly the set of the N zeros xn is an ordered set (hence an N-vector). We trust the reader to understand these rather obvious facts and therefore hereafter we refrain from any additional discussion of these issues. Our task now is to identify—for the special class of nongeneric polynomials (5)— equivalent relations to the identities (3), to be then used in order to identify new solvable dynamical systems. The first step is to time-differentiate once the formula (5), getting the relations

M M−m PM,t (z; t)= y˙m (t) z m=1 X N N µn−1 µℓ = − µnx˙ n (t)[z − xn (t)] {[z − xℓ (t)] } . (7) n=1 6 X ℓ=1Y, ℓ=n Remark 2-2. Hereafter, in order to avoid clattering our presentation with unessential details, we occasionally make the convenient assumption that all the numbers µn be different among themselves; the diligent reader shall have no difficulty to understand how the treatment can be extended to include cases in which this simplifying assumption does not hold—indeed in the specific examples discussed below we will include in our treatment also cases in which this simplification is not valid, taking appropriate care of such cases. And we

5 also assume—without loss of generality—that the numbers µn are ordered in decreasing order, µn ≥ µn+1. Our next step is to z-diﬀerentiate µ times the above formulas, ﬁrstly with µ =0, 1, 2, ..., µn −2 and secondly with µ = µn −1; and then set z = xn (for each value of n =1, 2, ..., N). There clearly thereby obtain the following formulas:

M−µ (M − m)! y˙ (t) [x (t)]M−m−µ =0 , m (M − m − µ)! n m=1 X µ = 0, 1, ..., µn − 2 , n =1, 2, ..., N ; (8a)

M−µ +1 n (M − m)! y˙ (t) [x (t)]M−m−µn+1 m (M − m − µ + 1)! n m=1 n X N µℓ = − (µn!)x ˙ n (t) {[xn (t) − xℓ (t)] } , 6 ℓ=1Y, ℓ=n n = 1, 2, ..., N . (8b)

The second set, (8b), yields the following N expressions of the time-derivatives of the N zeros xn (t) in terms of the time-derivatives of the M coeﬃcients ym (t):

−1 N x˙ (t)= − µ ! [x (t) − x (t)]µℓ · n  n n ℓ  6  ℓ=1Y, ℓ=n  M−µn+1  (M − m)!  M−m−µn+1 · y˙ (t) [x (t)] , m (M − m − µ + 1)! n m=1 n X n =1, 2, ..., N . (9)

The ﬁrst set, (8a), consists of linear relations among the M time-derivatives y˙m (t) of the M coeﬃcients ym (t): and it is easily seen, via (5b), that there are altogether N

(µm − 1) = M − N (10) n=1 X such relations. So one can select N quantitiesy ˙m (t)—let us hereafter call themy ˙m˜ (t)—and compute, from the M − N linear equations (8a), all the other M − N quantitiesy ˙m (t) with m 6=m ˜ as linear expressions in terms of these selected N quantitiesy ˙m˜ (t). The goal of expressing the N time-derivativesx ˙ n as linear equations—somehow analogous to the identities (3)—in terms of the N time-derivativesy ˙m˜ (t) of N, arbitrarily selected, coeﬃcients ym˜ (t) is thereby ﬁnally achieved. Indeed the task of expressing the M −N quantitiesy ˙m (t) with m 6=m ˜ in terms of the N quantitiesy ˙m˜ (t)—and of course the N zeros xn (t)— can in principle be implemented explicitly as it amounts to solving the M − N

6 linear equations (8) for the M − N unknownsy ˙m (t), with the N quantities y˙m˜ (t) playing there the role of known quantities; clearly implying that the resulting expressions of the M − N quantitiesy ˙m (t) are linear functions of the N quantitiesy ˙m˜ (t). And the insertion of these linear expressions of the M − N quantitiesy ˙m (t) (with m 6=m ˜ ) in terms of the N quantitiesy ˙m˜ (t) in the N formulas (9) fulfils our goal. The actual implementation of this development must of course be performed on a case-by-case basis, see below. In the special case with only one multiple zero—and if moreover the indicesm ˜ are assigned their first N values, i. e. m˜ = 1, 2, ..., N—these results shall reproduce the results of the path-breaking paper [11], which was confined to the treatment of this special case. The outcomes of these developments are detailed, in the special case with N = 2 and M = 4, in the following Section 3; for the motivation of this drastic restriction see below.

3 The N =2, M =4 case

In the special case with N = 2 the formula (9) simpliﬁes, reading (see (5b))

1+µ2 − (µ + µ − m)! x˙ = − [µ ! (x − x )µ2 ] 1 y˙ 1 2 (x )µ2−m+1 , (11a) 1 1 1 2 m (µ − m + 1)! 1 m=1 2 X 1+µ1 − (µ + µ − m)! x˙ = − [µ ! (x − x )µ1 ] 1 y˙ 1 2 (x )µ1−m+1 . (11b) 2 2 2 1 m (µ − m + 1)! 2 m=1 1 X Let us moreover restrict attention to the case with M =4, as the case with M = 3 (implying µ1 =2,µ2 = 1: see Remark 2-2) has been already discussed in [10] and [12] and the case with M = 4 is suﬃciently rich (see below) to deserve a full paper. In the case with M = 4 there are 2 possible assignments of the 2 parameters µn: (i) µ1 = 3, µ2 = 1; (ii) µ1 = µ2 = 2 (see Remark 2-2, and note that we now include also the case with µ1 = µ2).

3.1 Case (i): µ1 =3, µ2 =1 In this case (5a) clearly implies the following expressions of the 4 coeﬃcients ym (t) in terms of the 2 zeros xn (t):

y1 = − (3x1 + x2) , y2 =3x1 (x1 + x2) , 2 3 y3 = − (x1) (x1 +3x2) , y4 = (x1) x2 . (12)

Remark 3.1-1. Note that these formulas imply that x1 and x2 can be computed (in fact, explicitly!) from ym1 and ym2 —with m1 =1, 2, 3 and m1 < m2 =2, 3, 4) by solving an algebraic equation of degree m2.

7 The corresponding equations (8a) read

3 2 y˙1 (x1) +y ˙2 (x1) +y ˙3x1 +y ˙4 =0 , (13a)

2 3y ˙1 (x1) + 2y ˙2x1 +y ˙3 = 0 (13b) (note that in this case only the formulas with n = 1 are present). And the formulas (11) read

3 2 3x1y˙1 +y ˙2 (x2) y˙1 + (x2) y˙2 + x2y˙3 +y ˙4 x˙ 1 = − , x˙ 2 = 3 . (14) 3 (x1 − x2) (x1 − x2) There are now 6 possible diﬀerent assignments for the indicesm ˜ :

m˜ =1, 2;m ˜ =1, 3;m ˜ =1, 4;m ˜ =2, 3;m ˜ =2, 4;m ˜ =3, 4 , (15a) to which there correspond the 6 complementary assignments

m =3, 4; m =2, 4; m =2, 3; m =1, 4; m =1, 3; m =1, 2 . (15b)

Let us list below the 6 corresponding versions of the ODEs (14):

3x1y˙1 +y ˙2 (2x1 + x2)y ˙1 +y ˙2 x˙ 1 = − , x˙ 2 = , (16a) 3 (x1 − x2) (x1 − x2)

2 3 (x1) y˙1 − y˙3 x1 (x1 +2x2)y ˙1 − y˙3 x˙ 1 = − , x˙ 2 = , (16b) 6x1 (x1 − x2) 2x1 (x1 − x2) 3 2 (x1) y˙1 +y ˙4 (x1) x2y˙1 +y ˙4 x˙ 1 = − 2 , x˙ 2 = 2 , (16c) 3 (x1) (x1 − x2) (x1) (x1 − x2)

x1y˙2 +y ˙3 x1 (x1 +2x2)y ˙2 + (2x1 + x2)y ˙3 x˙ 1 = , x˙ 2 = − 2 , (16d) 3x1 (x1 − x2) 3 (x1) (x1 − x2) 2 2 (x1) y˙2 − 3y ˙4 (x1) x2y˙2 − (2x1 + x2)y ˙4 x˙ 1 = 2 , x˙ 2 = − 3 , (16e) 6 (x1) (x1 − x2) 2 (x1) (x1 − x2)

x1y˙3 + 3y ˙4 x1x2y˙3 + (x1 +2x2)y ˙4 x˙ 1 = − 2 , x˙ 2 = 3 . (16f) 3 (x1) (x1 − x2) (x1) (x1 − x2) Next, let us focus to begin with—in order to explain our approach—on the ﬁrst, (16a), of the 6 formulas (16). Assume moreover that the 2 quantities y1 (t) and y2 (t) evolve according to the following solvable system of ODEs:

y˙1 = f1 (y1,y2) , y˙2 = f2 (y1,y2) . (17)

It is then clear—via the identities (12)—that we can conclude that the dynamical system

−1 x˙ 1 = − [3 (x1 − x2)] [3x1f1 (− (3x1 + x2) , 3x1 (x1 + x2))

+f2 (− (3x1 + x2) , 3x1 (x1 + x2))] , (18a)

8 −1 x˙ 2 = (x1 − x2) [(2x1 + x2) f1 (− (3x1 + x2) , 3x1 (x1 + x2))

+f2 (− (3x1 + x2) , 3x1 (x1 + x2))] , (18b) is as well solvable. While this is in itself an interesting result—to become more signiﬁcant for explicit assignments of the 2 functions f1 (y1,y2) and f2 (y1,y2) (see below)—an additional interesting development emerges if—following the approach of [12]— we now assume the 2 functions f1 (y1,y2) and f2 (y1,y2) to be both polynomial in their 2 arguments and moreover such that 2 2 3xf1 −4x, 6x + f2 −4x, 6x =0; (19) a restriction that is clearly suﬃcient to guarantee that the right-hand sides of the equations of motion (21) become polynomials in the 2 dependent variables x1 (t) and x2 (t) (since the numerators in the right-hand sides of the 2 ODEs (21) are then both polynomials in the variables x1 and x2 which vanish when x1 = x2 = x and which therefore contain the factor x1 − x2). An representative example of such functions is 3 f1 (y1,y2)= α0 + α1y2 , f2 (y1,y2)= β0y1 + β1 (y1) , (20a) with (see (19)) 4β 32β α = 0 , α = 1 ; (20b) 0 3 1 9 note that the corresponding equations of motion (17) are then indeed solvable, see—up to trivial rescalings of some parameters—the solution in terms of Jaco- bian elliptic functions in Example 1 in [12], and, below, in Subsection Case A.3.1 of Appendix A. The conclusion is then that the dynamical system

2 2 x˙ 1 = a + b 5 (x1) + 10x1x2 + (x2) , (21a) h i 2 2 x˙ 2 = a + b 17 (x1) +2x1x2 − 3 (x2) , (21b) h i with a = −β0/3 and b = −β1/3 two arbitrary parameters, is solvable: indeed the solution of its initial-values problem—to evaluate x1 (t) and x2 (t) from arbitrarily assigned initial values x1 (0) and x2 (0)—are (explicitly!) yielded by the solution of a quadratic algebraic equation the coeﬃcients of which involve the Jacobian elliptic function µ sn(λt + ρ, k) with the 4 parameters µ, λ, ρ, k given by simple formulas in terms of the 2 initial data x1 (0) and x2 (0) and the 2 a priori arbitrary parameters a and b. (The interested reader can easily obtain all the relevant formulas from the treatment given above, comparing it if need be with the analogous treatment provided in Example 1 of [12]; or see below Subsection 4.7). Remark 3.1-2. An equivalent—indeed more direct—way to identify the solvable dynamical system (21) as corresponding to the solvable dynamical system 4 32β y˙ = β − 1 y , y˙ = β y + β (y )3 (22) 1 3 0 9 2 2 0 1 1 1

9 (see (17) and (20)), is via the relations

y1 = − (3x1 + x2) , y2 =3x1 (x1 + x2) (23a)

(see (12)) and their time derivatives,

y˙1 = − (3x ˙ 1 +x ˙ 2) , y˙2 = 3 [(2x1 + x2)x ˙ 1 + x1x˙ 2] . (23b)

3.2 Case (ii): µ1 = µ2 =2 In this case

2 2 y1 = −2 (x1 + x2) , y2 = (x1) + (x2) +4x1x2 , 2 y3 = −2x1x2 (x1 + x2) , y4 = (x1x2) . (24a)

Remark 3.2-1. Of course a remark completely analogous to Remark 3.1-1 holds in this case as well. The corresponding equations (8a) read

3 2 y˙1 (xn) +y ˙2 (xn) +y ˙3xn +y ˙4 =0 , n =1, 2 ; (24b) and proceeding as above one easily obtains, for the 6 assignments (15), the following systems of 2 ODEs:

(2x1 + x2)y ˙1 +y ˙2 (x1 +2x2)y ˙1 +y ˙2 x˙ 1 = − , x˙ 2 = , (25a) 2 (x1 − x2) 2 (x1 − x2)

x1 (x1 +2x2)y ˙1 − y˙3 x2 (x2 +2x1)y ˙1 +y ˙3 x˙ 1 = − , x˙ 2 = , (25b) 2 2 2 2 2 (x1) − (x2) 2 (x1) − (x2)

h 2 i h 2 i (x1) x2y˙1 +y ˙4 x1 (x2) y˙1 +y ˙4 x˙ 1 = − , x˙ 2 = , (25c) 2x1x2 (x1 − x2) 2x1x2 (x1 − x2)

x1 (x1 +2x2)y ˙2 + (2x1 + x2)y ˙3 x2 (x2 +2x1)y ˙2 + (2x2 + x1)y ˙3 x˙ 1 = , x˙ 2 = − , 3 3 3 3 2 (x1) − (x2) 2 (x1) − (x2) h i h i (25d) 2 2 (x1) x2y˙2 − (2x1 + x2)y ˙4 (x2) x1y˙2 − (2x2 + x1)y ˙4 x˙ 1 = , x˙ 2 = − (25e) 2 2 2 2 2x1x2 (x1) − (x2) 2x1x2 (x1) − (x2) h i h i

x1x2y˙3 + (x1 +2x2)y ˙4 x1x2y˙3 + (x2 +2x1)y ˙4 x˙ 1 = − 2 , x˙ 2 = 2 . (25f) 2x1 (x2) (x1 − x2) 2x2 (x1) (x1 − x2)

10 Hence, to the system (17), one now associates again the requirement (19); and—by making again the assignment (20a) for the system of evolution equations satisﬁed by y1 (t) and y2 (t)—one identiﬁes again the restriction (20b), thereby concluding—via (25a)—that the polynomial system

2 2 x˙ n = a + b (xn) − 8xnxn+1 − 5 (xn+1) , n =1, 2 mod [2] , (26) h i where now a = −β0/3 and b = 4β1/9, is solvable. And the explicit solution is then quite analogous (up to simple modiﬁcations of some parameters) to that described (after eq. (21)) in the preceding Subsection 3.1.

4 Other solvable systems of 2 nonlinearly-coupled ODEs identiﬁed via the technique described in Section 3

In this Section 4 we report a list of solvable systems of 2 nonlinearly coupled first-order ODEs satisfied by the 2 dependent variables x1 (t) and x2 (t); in each case we identify the corresponding solvable system of 2 ODEs satisfied by 2 variables ym˜ (t) (for these, and other, notations used below see Section 3); indeed, to help the reader mainly interested in the solvable character of one of the following systems we also specify below on a case-by-case basis the information which allows to solve that specific system (we do so even at the cost of minor repetitions). Note that the majority of these models feature equations of type (4), but in a few cases the right-hand sides of these ODEs are not quite polynomial. And let us recall that in this Section 4 parameters such as a, b, c (possibly equipped with indices) are arbitrary numbers (possibly complex). Remark 4-1. Most of the models reported below are characterized by evolution equations of the following kind:

K (n) x˙ n = pk (x1, x2) , n =1, 2 , (27a) Xk=0 h i (n) with K a positive integer and the functions pk (x1, x2) homogenous polynomials of degree k,

k (n) (n,k) k−ℓ ℓ pk (x1, x2)= aℓ (x1) (x2) , k =0, 1, ..., K , n =1, 2 . (27b) Xℓ=0 h i So the diﬀerent models are characterized by the assignments of the positive 2 (n,k) integer K and of the (K + 1) parameters aℓ , expressed in each case in terms of a few arbitrary parameters. It is of course obvious that in all the models associated with Case (ii) (see Subsection 3.2) these parameters satisfy the (1,k) (2,k) restriction aℓ = aℓ , since in that case the 2 zeros x1 (t) and x2 (t) are

11 completely equivalent; while this restriction need not hold in Case (i) (see Subsection 3.1), although in some such cases it also emerges (see below). It is on the other hand plain that, also in Case (i) (as, obviously, in Case (ii)), there holds the restriction k k (1,k) (2,k) aℓ = aℓ , k =0, 1, ..., K , (27c) Xℓ=0 h i Xℓ=0 h i because for the special initial conditions x1 (0) = x2 (0)—implying x1 (t) = x2 (t) ≡ x (t), since in such case the distinction among Case (i) and Case (ii) obviously disappears—the 2 evolution equations (with n =1, 2) satisfied by x (t), k k (n,k) x˙ = x aℓ , k =0, 1, ..., K , n =1, 2 , (27d) Xℓ=0 h i must coincide. Remark 4-2. In the following 44 subsections we list as many solvable systems of 2 nonlinearly-coupled first-order ODEs, most of them with polynomial right-hand sides, and we indicate how each of them can be solved. The presentation of all these models is made so as to facilitate the utilization of these findings by practitioners only interested in one of these models (or its generalization, see Section 5). Note however that not all these models are different among themselves: indeed, some feature identical equations of motion— although the method to solve them might seem different. This is demonstrated by the following self-evident identification of the following equations of motion: (32)≡(54), (33)≡(55), (40)≡(56)≡(70), (41)≡(57)≡(71), (48)≡(62)≡(68), (49)≡(63)≡(69). So in fact the list below contains only 34 different systems of 2 nonlinearly-coupled first-order differential equations for the 2 time-dependent variables x1 (t) and x2 (t).

4.1 Model 4.(i)1.2a

2 2 x˙ 1 = x1 a + b 11 (x1) +6x1x2 − (x2) , (28a)

n 3 h 2 2 io 3 x˙ 2 = ax2 + b −6 (x1) + 9 (x1) x2 + 12x1 (x2) + (x2) ; (28b) h i x1 (t) and x2 (t) are related to y1 (t) and y2 (t) by (12); and the variables y1 (t) and y2 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =1, m˜ 2 =2,L =1, α0 = a, α1 =3b, β0 =2a, β1 = 16b. 4.2 Model 4.(ii)1.2a

3 2 3 x˙ n = axn + b 4 (xn) + 9 (xn) xn+1 − (xn+1) , n =1, 2 mod[2] ; (29) h i 12 x1 (t) and x2 (t) are related to y1 (t) and y2 (t) by (24a); and the variables y1 (t) and y2 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =1, m˜ 2 =2,L =1, α0 = a, α1 = (3/4) b, β0 =2a, β1 =4b. 4.3 Model 4.(i)1.2b

2 2 3 x˙ n = a0 + xn a1 + a2X + a3X + b1X + b2X + b3X , X ≡ 3x1 + x2, n =1, 2; (30a) x1 (t) and x2 (t) are related to y1 (t) and y2 (t) by (12); and the variables y1 (t) and y2 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =1, m˜ 2 = 2,L = 3, α0 = −4a0, α1 = a1 +4b1, α2 = − (a2 +4b2) , α3 = a3 +4b3, β1 =2a1, β2 = −2a2, β3 =2a3, γ0 = −3a0, γ1 =3b1, γ2 = −3b2, γ3 =3b3. This model (30a) (with a0 = 0) is actually a special case of the more general model L ℓ−1 x˙ n = a0 + (−X) [aℓxn + bℓX] , X ≡ 3x1 + x2, n =1, 2 , (30b) Xℓ=1 n o again with x1 (t) and x2 (t) related to y1 (t) and y2 (t) by (12) and the variables y1 (t) and y2 (t) evolving according to (87) withm ˜ 1 =1, m˜ 2 =2,L an arbitrary positive integer, α0 = −4a0, αℓ = aℓ +4bℓ, βℓ =2aℓ, γ0 = −3a0, γℓ =3bℓ.

4.4 Model 4.(ii)1.2b

2 2 3 x˙ n = a0 + xn a1 + a2X + a3X + b1X + b2X + b3X , X ≡ x1 + x2 , n =1, 2; (31a) x1 (t) and x2 (t) are related to y1 (t) and y2 (t) by (24a); and the variables y1 (t) and y2 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =1, m˜ 2 =2,L =3, α0 = −4a0, α1 = a1 +2b1, α2 = −a2/2 − b2, α3 = b3/2+ a3/4, β1 = 2a1, β2 = −a2, β3 = a3/2, γ0 = −3a0, γ1 = (3/2) b1, γ2 = − (3/4) b2, γ3 = (3/8) b3. This model (31a) is actually a special case of the more general model

L ℓ−1 x˙ n = a0 + (−2X) [aℓxn + bℓX] , X ≡ x1 + x2 , n =1, 2 mod [2] , ℓ=1 n o X (31b) again with x1 (t) and x2 (t) related to y1 (t) and y2 (t) by (24a) and the variables y1 (t) and y2 (t) evolving according to (87) withm ˜ 1 =1, m˜ 2 =2,L an arbitrary positive integer, α0 = −4a0, αℓ = aℓ +2bℓ, βℓ =2aℓ, γ0 = −3a0, γℓ = (3/2) bℓ.

13 4.5 Model 4.(i)1.2c 2 x˙ n = xn a + bX + cX , X ≡ x1 (x1 + x2) , n =1, 2 ; (32) x1 (t) and x2 (t) are related to y1 (t) and y2 (t) by (12); and the variables y1 (t) and y2 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =2, m˜ 2 = 1,L = 3, α0 = 0, α1 = 2a, α2 = 2b/3, α3 = 2c/9, β1 = a, β2 = b/3, β3 = c/9, γℓ =0.

4.6 Model 4.(ii)1.2c

2 x˙ n = xn a + bX + cX , 2 2 X ≡ (x1) +4x1x2 + (x2) , n =1, 2 ; (33) x1 (t) and x2 (t) are related to y1 (t) and y2 (t) by (24a); and the variables y1 (t) and y2 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =2, m˜ 2 = 1,L = 3, α0 = 0, α1 = 2a, α2 = 2b, α3 = 2c, β1 = a, β2 = b, β3 = c, γℓ =0.

4.7 Model 4.(i)1.2d

2 2 x˙ 1 = a + b 5 (x1) + 10x1x2 + (x2) , (34a)

h 2 2i x˙ 2 = a + b 17 (x1) +2x1x2 − 3 (x2) ; (34b) h i x1 (t) and x2 (t) are related to y1 (t) and y2 (t) by (12); and the variables y1 (t) and y2 (t) evolve according to (95a), the explicit solution of which is given by the relevant formulas in Subsection Case A.3.1 of Appendix A withm ˜ 1 = 1, m˜ 2 =2, α0 = −4a, α1 = − (32/3) b, β0 = −3a, β1 = −3b. Note that this is the model treated in detail in Subsection 3.1.1, see (21).

4.8 Model 4.(ii)1.2d

2 2 x˙ n = a + b (xn) − 8x1x2 − 5 (xn+1) , n =1, 2 mod[2] ; (35) h i x1 (t) and x2 (t) are related to y1 (t) and y2 (t) by (24a); and the variables y1 (t) and y2 (t) evolve according to (95a), the explicit solution of which is given by the relevant formulas in Subsection Case A.3.1 of Appendix A withm ˜ 1 = 1, m˜ 2 = 2, α0 = −4a, α1 = 8b, β0 = −3a, β1 = (9/4) b. Note that this is the model treated in detail in Subsection 3.1.2, see (26).

14 4.9 Model 4.(i)1.3a

3 2 2 3 x˙ 1 = x1 a + b 65 (x1) +77(x1) x2 − 13x1 (x2) − (x2) , (36a)

n h4 3 2 2 io3 4 x˙ 2 = ax2 − b 33 (x1) +15(x1) x2 − 147(x1) (x2) − 27x1 (x2) − 2 (x2) ; h (36b)i x1 (t) and x2 (t) are related to y1 (t) and y3 (t) by (12); and the variables y1 (t) and y3 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A) withm ˜ 1 = 1, m˜ 2 =3,L =1, α0 = a, α1 = −2b, β0 =3a, β1 = −96b.

4.10 Model 4.(ii)1.3a

2 2 x˙ n = xn a + b (x1 + x2) (xn) +5x1x2 − 2 (xn+1) , n =1, 2 mod [2] ; n h io (37) x1 (t) and x2 (t) are related to y1 (t) and y3 (t) by (24a); and the variables y1 (t) and y3 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =1, m˜ 2 =3,L =1, α0 = a, α1 = −b/8, β0 =3a, β1 = −6b.

4.11 Model 4.(i)1.3b

−1 2 2 x˙ 1 = (6x1) (x1) a0 + a1X + a2X h 2 3 + (7x1 + x2) b0 + b1X + b2X + b3X , (38a) −1 2 x˙ 2 = (6x1) x1x2 a0 + a1X + a2X 2 3 +(11x1 − 3x2) b0 + b1X + b2X + b3X , (38b) X ≡ 3x1 + x2 ; (38c) x1 (t) and x2 (t) are related to y1 (t) and y3 (t) by (12); and the variables y1 (t) and y3 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A with ℓ−1 m˜ 1 = 1, m˜ 2 = 3,L = 3, α0 = − (16/3) b0, αℓ = (−1) (aℓ−1/6) + (16/3) bℓ, ℓ−1 ℓ βℓ = (−1) aℓ−1/2, (ℓ =1, 2, 3) , γℓ−1 = (−1) bℓ−1, (ℓ =1, 2, 3, 4) . Note that the right-hand sides of these 2 ODEs, (38), are both polynomial only if the 4 parameters bℓ vanish, bℓ =0, ℓ =0, 1, 2, 3.

15 4.12 Model 4.(ii)1.3b

−1 2 x˙ 1 =6 xn a1 + a2X + a3X −1 2 + (xn +3xn+1 ) b0X + b1 + b2X + b3X , X ≡ x1 + x2 , n =1, 2mod[2]; (39) x1 (t) and x2 (t) are related to y1 (t) and y3 (t) by (24a); and the variables y1 (t) and y3 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =1, m˜ 2 = −ℓ 5−ℓ −ℓ−1 3,L =3, α0 = − (16/3) b0, αℓ = − (−2) aℓ +2 bℓ /3,βℓ = − (−2) aℓ, −ℓ γ0 = −b0, γℓ = − (−2) bℓ, ℓ =1, 2, 3. Note that the right-hand sides of these 2 ODEs, (38), are both polynomial iﬀ the single parameter b0 vanishes, b0 = 0.

4.13 Model 4.(i)1.3c

2 2 x˙ n = xn a + bX + cX , X ≡ (x1) (x1 +3x2) , n =1, 2 ; (40) x1 (t) and x2 (t) are related to y1 (t) and y3 (t) by (12); and the variables y1 (t) and y3 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =3, m˜ 2 =1,L =3, α0 =0, α1 =3a, α2 = −3b, α3 =3c, β1 = a, β2 = −b, β3 = c, γℓ =0.

4.14 Model 4.(ii)1.3c 2 x˙ n = xn a + bX + cX ,X ≡ x1x2 (x1 + x2) , n =1, 2 ; (41) x1 (t) and x2 (t) are related to y1 (t) and y3 (t) by (24a); and the variables y1 (t) and y3 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A) withm ˜ 1 = 3, m˜ 2 = 1,L = 3, α0 = 0, α1 = 3a, α2 = −3b/2, α3 = 3c/4, β1 = a, β2 = −b/2, β3 = c/4, γℓ =0.

4.15 Model 4.(i)1.3d

−1 x˙ 1 = (6x1) {a (7x1 + x2) 4 3 2 3 4 +b 13 (x1) +376(x1) x2 +106(x1x2) + 16x1 (x2) + (x2) , (42a) h io

−1 x˙ 2 = (6x1) {a (11x1 − 3x2) 4 3 2 3 4 +b 473(x1) +408(x1) x2 − 318(x1x2) − 48x1 (x2) − 3 (x2) ; (42b) h io 16 x1 (t) and x2 (t) are related to y1 (t) and y3 (t) by (12); and the variables y1 (t) and y3 (t) evolve according to (95a), the explicit solution of which is given by the relevant formulas in Subsection Case A.3.2 of Appendix A withm ˜ 1 = 1, 2 m˜ 2 = 3, α0 = − (16/3) a, α1 = 16 /3 b, β0 = −a, β1 = b. Note that the right-hand sides of the 2 ODEs (42) are not polynomial.

4.16 Model 4.(ii)1.3d

x +3x x˙ = a n n+1 1 x + x 1 2 3 2 2 3 +b 3 (xn) − (xn) xn+1 − 15xn (xn+1) − 3 (xn+1) , n =1h , 2 mod [2] ; i (43) x1 (t) and x2 (t) are related to y1 (t) and y3 (t) by (24a); and the variables y1 (t) and y3 (t) evolve according to (95a), the explicit solution of which is given by the relevant formulas in Subsection Case A.3.2 of Appendix A withm ˜ 1 = 1, m˜ 2 =3, α0 = −8a, α1 = −16b, β0 = − (3/2) a, β1 = − (3/16) b. Note that the right-hand sides of the 2 ODEs (43) are polynomial only if a =0.

4.17 Model 4.(i)1.4a

x˙ 1 = x1 {a + (b/3) · 4 3 2 3 4 · 243(x1) +648(x1) x2 − 106(x1x2) − 16x1 (x2) − (x2) , (44a) h io

x˙ 2 = x2 {a − b· 4 3 2 3 4 · 243(x1) − 376(x1) x2 − 106(x1x2) − 16x1 (x2) − (x2) ; (44b) h io x1 (t) and x2 (t) are related to y1 (t) and y4 (t) by (12); and the variables y1 (t) and y4 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =1, 10 m˜ 2 =4,L =1, α0 = a, α1 = b, β0 =4a, β1 =2 b = 1024b.

4.18 Model 4.(ii)1.4a

x˙ n = xn {a + b· 4 3 2 3 4 · (xn) + 6 (xn) xn+1 +16(x1x2) − 6xn (xn+1) − (xn+1) , nh=1, 2 mod [2] , io (45)

17 x1 (t) and x2 (t) are related to y1 (t) and y4 (t) by (24a); and the variables y1 (t) and y4 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =1, 6 m˜ 2 =4,L =1, α0 = a, α1 = b/16,β0 =4a, β1 =2 b = 64b.

4.19 Model 4.(i)1.4b

2 x˙ 1 = x1 a0 + a1X + a2X 2 2 37 (x1) + 10x1x2 + (x2) 2 3 − 2 b0 + b1X + b2X + b3X , (46a) " 3 (x1) #

2 x˙ 2 = x2 a0 + a1X + a2X 2 2 27 (x1) − 10x1x2 − (x2) 2 3 − 2 b0 + b1X + b2X + b3X ; (46b) " (x1) # X ≡ 3x1 + x2 ; (46c) x1 (t) and x2 (t) are related to y1 (t) and y4 (t) by (12); and the variables y1 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =1, ℓ−1 ℓ−1 m˜ 2 = 4,L = 3, α0 = 64b0; αℓ = (−1) (aℓ−1 − 64bℓ) , βℓ = (−1) 4aℓ−1, ℓ ℓ =1, 2, 3; γℓ = (−1) bℓ, ℓ =0, 1, 2, 3. Note that the right-hand sides of these 2 ODEs, (46), are both polynomial only if all the 4 parameters bℓ vanish, bℓ =0, ℓ =0, 1, 2, 3.

4.20 Model 4.(ii)1.4b

2 2 2 (xn) − 4x1x2 − (xn+1) x˙ n = xn a0 + a1X + a2X + · " x1x2 # 2 3 · b0 + b1X + b2X + b3X , X ≡ x1 + x2 , n =1, 2 mod[2] ; (47) x1 (t) and x2 (t) are related to y1 (t) and y4 (t) by (24a); and the variables y1 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =1, 1−ℓ 4−ℓ 3−ℓ m˜ 2 =4,L =3, α0 = 16b0; αℓ = (−2) aℓ−1 + (−2) bℓ, βℓ = (−2) aℓ−1, −2−ℓ ℓ = 1, 2, 3; γℓ = (−2) bℓ, ℓ = 0, 1, 2, 3. Note that the right-hand sides of these 2 ODEs, (47), are both polynomial only if all the 4 parameters bℓ vanish, bℓ =0, ℓ =1, 2, 3.

18 4.21 Model 4.(i)1.4c

2 3 x˙ n = xn a + bX + cX , X ≡ (x1) x2 , n =1, 2 ; (48) x1 (t) and x2 (t) are related to y1 (t) and y4 (t) by (12); and the variables y1 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =4, m˜ 2 = 1,L = 3, α0 = 0, α1 = 4a, α2 = 4b, α3 = 4c, β1 = a, β2 = b, β3 = c, γℓ =0.

4.22 Model 4.(ii)1.4c

2 2 x˙ n = xn a + bX + cX , X ≡ (x1x2) , n =1, 2 ; (49) x1 (t) and x2 (t) are related to y1 (t) and y4 (t) by (24a); and the variables y1 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =4, m˜ 2 = 1,L = 3, α0 = 0, α1 = 4a, α2 = 4b, α3 = 4c, β1 = a, β2 = b, β3 = c, γℓ =0.

4.23 Model 4.(i)1.4d

−1 2 2 2 x˙ 1 = 3 (x1) a 37 (x1) + 10x1x2 + (x2)

h i6 n h 5 4 i2 3 +b 2187 (x1) − 9094 (x1) x2 − 3991 (x1) (x2) − 1156 (x1x2)

h 2 4 5 6 −211(x1) (x2) − 22x1 (x2) − (x2) , (50a) io

−1 2 2 2 x˙ 2 = (x1) a 27 (x1) − 10x1x2 − (x2)

h i 6 n h 5 4 i 2 3 −b 2187 (x1) + 7290(x1) x2 − 3991 (x1) (x2) − 1156 (x1x2)

h 2 4 5 6 −211(x1) (x2) − 22x1 (x2) − (x2) ; (50b) io x1 (t) and x2 (t) are related to y1 (t) and y4 (t) by (12); and the variables y1 (t) and y4 (t) evolve according to (95a), the explicit solution of which is given by the relevant formulas in Subsection Case A.3.3 of Appendix A withm ˜ 1 = 1, 6 14 m˜ 2 = 4, α0 = −2 a = −64a; α1 = 2 b = 1638b, β0 = −a, β1 = b. Note that the right-hand sides of these 2 ODEs, (50), are not polynomial.

19 4.24 Model 4.(ii)1.4d

−1 2 2 x˙ n = (x1x2) a (xn) − 4x1x2 − (xn+1)

6 n h5 4 2 i 3 +b (xn) + 8 (xn) xn+1 +29(xn) (xn+1) − 64 (x1x2)

h 2 4 5 6 −29 (xn) (xn+1) − 8xn (xn+1) − (xn+1) , n =1, 2 mod [2] ; io (51) x1 (t) and x2 (t) are related to y1 (t) and y4 (t) by (24a); and the variables y1 (t) and y4 (t) evolve according to (95a), the explicit solution of which is given by the relevant formulas in Subsection Case A.3.3 of Appendix A withm ˜ 1 = 1, 4 8 −2 −6 m˜ 2 =4, α0 =2 a = 16a; α1 =2 b = 256b, β0 =2 a = a/4,β1 =2 b = b/64. Note that the right-hand sides of these 2 ODEs, (50), are not polynomial.

4.25 Model 4.(i)2.3a

3 3 2 2 3 x˙ 1 = x1 a + b (x1) 3 (x1) +10(x1) x2 +7x1 (x2) − 4 (x2) , (52a)

n 3 h 4 3 3 4 io x˙ 2 = ax2 − b (x1) 2 (x1) + 7 (x1) x2 − 17x1 (x2) − 8 (x2) ; (52b) h i x1 (t) and x2 (t) are related to y2 (t) and y3 (t) by (12); and the variables y2 (t) and y3 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =2, m˜ 2 =3,L =1, α0 =2a, α1 = (4/27) b, β0 =3a, β1 =3b.

4.26 Model 4.(ii)2.3a

−1 2 2 x˙ n = xn a + b (x1) + x1x2 + (x2) · 8 h 7 6 i 2 5 3 · (xn) +19(xn) xn+1 +151(xn) (xn+1) +331(xn) (xn+1) h 4 3 5 2 6 7 +259 (x1x2) +13(xn) (xn+1) − 89 (xn) (xn+1) − 35xn (xn+1) 8 −2 (xn+1) , n =1, 2mod[2]; (53) io x1 (t) and x2 (t) are related to y2 (t) and y3 (t) by (24a); and the variables y2 (t) and y3 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =2, m˜ 2 = 3,L = 1, α0 = 2a, α1 = 2b, β0 = 3a, β1 = (81/2) b. Note that the right-hand sides of these ODEs is not polynomial.

20 4.27 Model 4.(i)2.3b 2 x˙ n = xn a + bX + cX , X ≡ x1 (x1 + x2) , n =1, 2 ; (54) x1 (t) and x2 (t) are related to y2(t) and y3 (t) by (12); and the variables y2 (t) and y3 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =2, m˜ 2 = 3,L = 3, α0 = 0 , α1 = 2a, α2 = (2/3) b, α3 = (2/9) c, β1 = 3a, β2 = b, β3 = c/3, γℓ =0.

4.28 Model 4.(ii)2.3b

2 2 2 x˙ n = xn a + bX + cX , X ≡ (x1) +4x1x2 + (x2) , n =1, 2 ; (55) x1 (t) and x2 (t) are related to y2 (t) and y3 (t) by (24a); and the variables y2 (t) and y3 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =2, m˜ 2 = 3,L = 3, α0 = 0 , α1 = 2a, α2 = 2b, α3 = 2c, β1 = 3a, β2 = 3b, β3 = 3c, γℓ =0.

4.29 Model 4.(i)2.3c

2 2 x˙ n = xn a + bX + cX , X ≡ (x1) (x1 +3x2) , n =1, 2 ; (56) x1 (t) and x2 (t) are related to y2 (t) and y3 (t) by (12); and the variables y2 (t) and y3 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =3, m˜ 2 = 2,L = 3, α0 = 0 , α1 = 3a, α2 = −3b, α3 = 3c, β1 = 2a, β2 = −2b, β3 =3c, γℓ =0.

4.30 Model 4.(ii)2.3c 2 x˙ n = xn a + bX + cX , X ≡ x1x2 (x1 + x2) , n =1, 2 ; (57) x1 (t) and x2 (t) are related to y2(t) and y3 (t) by (24a); and the variables y2 (t) and y3 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =3, m˜ 2 = 2,L = 3, α0 = 0 , α1 = 3a, α2 = − (3/2) b, α3 = (3/4) c, β1 = 2a, β2 = −b, β3 = c/2, γℓ =0.

21 4.31 Model 4.(i)2.4a

2 2 2 x˙ 1 = x1 a + b (x1) (x1) +4x1x2 − (x2)

4n 4 h 3 2 i 3 4 +c (x1) (x1) + 6 (x1) x2 +16(x1x2) − 6x1 (x2) − (x2) ,(58a) h io

2 2 2 x˙ 2 = x2 a − b (x1) 3 (x1) − 4x1x2 − 3 (x2)

4n 4 h 3 2 i 3 4 +c (x1) −3 (x1) − 18 (x1) x2 +16(x1x2) + 18x1 (x2) + 3 (x2) , h io(58b) x1 (t) and x2 (t) are related to y2 (t) and y4 (t) by (12); and the variables y2 (t) and y4 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =2, m˜ 2 = 4,L = 2, α0 = 2a, α1 = (2/9) b, α2 = (2/81) c, β0 = 4a, β1 = 16b, β2 = 64c.

4.32 Model 4.(ii)2.4a

−1 5 4 3 2 x˙ n = xn a + (x1 + x2) · b (xn) +13(xn) xn+1 +64(xn) (xn+1)

2n 3 n h 4 5 +8 (xn) (xn+1) − 13xn (xn+1) − (xn+1)

9 8 7 i 2 6 3 +c (xn) +21(xn) xn+1 +186(xn) (xn+1) +906(xn) (xn+1) h 5 4 4 5 3 6 +2676 (xn) (xn+1) − 84 (xn) (xn+1) − 906(xn) (xn+1) 2 7 8 9 −186(xn) (xn+1) − 21xn (xn+1) − (xn+1) , n =1, 2 mod [2] ; ioo (59) x1 (t) and x2 (t) are related to y2 (t) and y4 (t) by (24a); and the variables y2 (t) and y4 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 = 2, m˜ 2 = 4,L = 2, α0 = 2a, α1 = 2b, α2 = −2c, β0 = 4a, β1 = 144b, 6 4 β2 = −2 3 c = −5184c. Note that the right-hand sides of these ODEs are not polynomial, except for the trivial case with b = c =0.

4.33 Model 4.(i)2.4b

2 −1 2 3 x˙ 1 = x1 a0 + a1X + a2X + (x1) b0 + b1X + b2X + b3X , (60a)

2 (2x1 − x2) 2 3 x˙ 2 = x2 a0 + a1X + a2X + 2 b0 + b1X + b2X + b3X , (60b) (x1) 22 X ≡ x1 (x1 + x2); (60c) x1 (t) and x2 (t) are related to y2 (t) and y4 (t) by (12); and the variables y2 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =2, 1−ℓ 1−ℓ 1−ℓ m˜ 2 =4,L =3, α0 = 12b0; αℓ =2 3 aℓ−1 +4 3 bℓ, βℓ =4 3 aℓ−1, ℓ = 1, 2, 3; γ = 2 3−1−ℓ b , ℓ = 0, 1, 2, 3. Note that the right-hand sides of ℓ ℓ these 2 ODEs, (46), are both polynomial only if all the 4 parameters b vanish, ℓ bℓ =0, ℓ =0, 1, 2, 3.

4.34 Model 4.(ii)2.4b

2 x˙ n = xn a0 + a1X + a2X 2 2 −2 (xn) +7xnxn+1 + (xn+1) 2 3 + b0 + b1X + b2X + b3X , " x1x2 (x1 + x2) # ) 2 2 X ≡ (x1) +4x1x2 + (x2) ; (61) x1 (t) and x2 (t) are related to y2 (t) and y4 (t) by (24a); and the variables y2 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =2, m˜ 2 =4,L =3, α0 = 36b0; αℓ =2aℓ−1 + 36bℓ, βℓ =4aℓ−1, ℓ =1, 2, 3; γℓ =2bℓ, ℓ = 0, 1, 2, 3. Note that the right-hand sides of these 2 ODEs, (61), are both polynomial only if all the 4 parameters bℓ vanish, bℓ =0, ℓ =0, 1, 2, 3.

4.35 Model 4.(i)2.4c

2 3 x˙ n = xn a + bX + cX , X ≡ (x1) x2 , n =1, 2 ; (62) x1 (t) and x2 (t) are related to y2 (t) and y4 (t) by (12); and the variables y2 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =4, m˜ 2 = 2,L = 3, α0 = 0 , α1 = 4a, α2 = 4b, α3 = 4c, β1 = 2a, β2 = 2b, β3 = 2c, γℓ =0.

4.36 Model 4.(ii)2.4c

2 2 x˙ n = xn a + bX + cX , X ≡ (x1x2) , n =1, 2 ; (63) x1 (t) and x2 (t) are related to y2 (t) and y4 (t) by (24a); and the variables y2 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =4, m˜ 2 = 2,L = 3, α0 = 0 , α1 = 4a, α2 = 4b, α3 = 4c, β1 = 2a, β2 = 2b, β3 = 2c, γℓ =0.

23 4.37 Model 4.(i)2.4d

−1 2 2 2 x˙ 1 = (x1) a + b (x1) (x1) − 4x1x2 − (x2) , (64a)

−2 n 2 h 3 2 io 2 3 x˙ 2 = (x1) a (2x1 − x2) − b (x1) 2 (x1) + 9 (x1) x2 − 6x1 (x2) − (x2) , n h (64b)io x1 (t) and x2 (t) are related to y2 (t) and y4 (t) by (12); and the variables y2 (t) and y4 (t) evolve according to (95a), the explicit solution of which is given by the relevant formulas in Subsection Case A.3.1 of Appendix A withm ˜ 1 = 2, m˜ 2 = 4, α0 = 12a; α1 = −48b, β0 = (2/3) a, β1 = − (2/27) b. Note that the right-hand sides of these 2 ODEs, (50), are not polynomial, unless a vanishes.

4.38 Model 4.(ii)2.4d

−1 2 2 x˙ n = [x1x2 (x1 + x2)] a 2 (xn) − 7x1x2 − (xn+1)

6 5 n h 4 2 i 3 +b 2 (xn) +27(xn) xn+1 +141(xn) (xn+1) − 280(x1x2)

h 2 4 5 6 −90 (xn) (xn+1) − 15xn (xn+1) − (xn+1) , n =1, 2 mod [2] ; io (65) x1 (t) and x2 (t) are related to y2 (t) and y4 (t) by (24a); and the variables y2 (t) and y4 (t) evolve according to (95a), the explicit solution of which is given by the relevant formulas in Subsection Case A.3.1 of Appendix A withm ˜ 1 = 2, m˜ 2 = 4, α0 = −36a; α1 = −1296b, β0 = −2a, β1 = −9b. Note that the right- hand sides of these 2 ODEs, (65), are not polynomial.

4.39 Model 4.(i)3.4a

8 x˙ 1 = x1 a + b (x1) ·

4 3 2 3 n 4 · (x1) +16(x1) x2 + 106(x1x2) + 376x1 (x2) − 243(x2) , (66a) h io 8 x˙ 2 = x2 a − b (x1) ·

4 3 2 3 n 4 · 3 (x1) +48(x1) x2 +318(x1x2) + 104x1 (x2) − 729(x2) ; (66b) h io x1 (t) and x2 (t) are related to y3 (t) and y4 (t) by (12); and the variables y3 (t) and y4 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =3, 10 m˜ 2 =4,L =1, α0 =3a, α1 =3b, β0 =4a, β1 =2 b = 1024b.

24 4.40 Model 4.(ii)3.4a

4 x˙ n = xn a + b (x1x2) ·

4 3 2 3 n 4 · 3 (xn) +18(xn) xn+1 +16(x1x2) − 18xn (xn+1) − 3 (xn+1) , h n =1, 2 mod[2];io (67) x1 (t) and x2 (t) are related to y3 (t) and y4 (t) by (24a); and the variables y3 (t) and y4 (t) evolve according to (82), the explicit solution of which is given by the relevant formulas in Subsection Case A.1 of Appendix A withm ˜ 1 =3, m˜ 2 =4,L =1, α0 =3a, α1 = (3/16) b, β0 =4a, β1 = 64b.

4.41 Model 4.(i)3.4b

2 3 x˙ n = xn a + bX + cX , X ≡ (x1) x2 , n =1, 2 ; (68) x1 (t) and x2 (t) are related to y3 (t) and y4 (t) by (12); and the variables y3 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =4, m˜ 2 = 3,L = 3, α0 = 0 , α1 = 4a, α2 = 4b, α3 = 4c, β1 = 3a, β2 = 3b, β3 = 3c, γℓ =0.

4.42 Model 4.(ii)3.4b

2 2 x˙ n = xn a + bX + cX , X = (x1x2) n =1, 2 ; (69) x1 (t) and x2 (t) are related to y3 (t) and y4 (t) by (24a); and the variables y3 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =4, m˜ 2 =3,L =3, α0 =0 , α1 =4a, α2 =4b, α3 =4c, β1 =3a, β2 =3b, β3 =3c, γℓ =0.

4.43 Model 4.(i)3.4c

2 2 x˙ n = xn a + bX + cX , X ≡ (x1) (x1 +3x2) , n =1, 2 ; (70) x1 (t) and x2 (t) are related to y3 (t) and y4 (t) by (12); and the variables y3 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =3, m˜ 2 =4,L =3, α0 =0 , α1 =3a, α2 = −3b, α3 =3c, β1 =4a, β2 = −4b, β3 = 4c, γℓ =0.

25 4.44 Model 4.(ii)3.4c 2 x˙ n = xn a + bX + cX , X ≡ x1x2 (x1 + x2) n =1, 2 ; (71) x1 (t) and x2 (t) are related to y3 (t) and y4 (t) by (24a); and the variables y3 (t) and y4 (t) evolve according to (87), the explicit solution of which is given by the relevant formulas in Subsection Case A.2 of Appendix A withm ˜ 1 =3, m˜ 2 = 4,L = 3, α0 = 0 , α1 = 3a, α2 = − (3/2) b, α3 = (3/4) c, β1 = 4a, β2 = −2b, β3 = c, γℓ =0.

5 Extensions

In this Section 5 we tersely indicate the possibility to generalize the class of solvable models listed in the preceding Section 4, by outlining the procedure to do so in just one case, that detailed in Subsection 4.3 (see (30a)), in fact just the special case of it with a0 = a1 = a3 = b1 = b3 = 0, so that its equations of motion read as follows:

x˙ n = (a2xn + b2X) X, X ≡ 3x1 + x2, n =1, 2 , (72a) namely 2 2 x˙ n = cn1 (x1) + cn2 (x2) + cn3x1x2 , n =1, 2 , (72b) with the 6 parameters cnm, n = 1, 2, m =1, 2, 3, expressed as follows in terms of the 2 a priori arbitrary parameters a2 and b2 (see (30a)):

c11 = 3 (a2 +3b2) , c12 = b2 , c13 = a2 +6b2 ,

c21 = 9b2 , c22 = a2 + b2 , c23 = 3 (a2 +2b2) . (72c)

The explicit solution of the initial-values problem of this system (72) is provided by Remark A.2-1 (see Subsection A.2 of Appendix A). Remark 5-1. Note that the right-hand sides of the 2 ODEs (72) are homogeneous polynomials of second degree, the coeﬃcients of which satisfy of course the condition 3 3 (c1ℓ)= (c2ℓ) , (73) Xℓ=1 Xℓ=1 as implied by Remark 4-1. Moreover—as clearly implied by (72a)—the 2 homogeneous second-degree polynomials in the right-hand sides of the 2 ODEs characterizing this model feature a common zero: they both vanish when X = 0, namely when x2 = −3x1. Analogous extensions of other models treated in this paper shall be performed by practitioners interested in these systems of ODEs in the context of speciﬁc applications (see Section 6).

26 Remark 5-2. Note that, via a by now well-known trick (see, for instance, [7]) corresponding to the following time-dependent change of both independent and dependent variables,

τ = exp(at); Xn (t) = exp(at) xn (τ) , n =1, 2 , (74) the autonomous system (72) gets replaced by the following, also autonomous, system:

2 2 X˙ n = aXn + cn1 (X1) + cn2 (X2) + cn3X1X2 , n =1, 2 . (75)

Here a is an arbitrary (time-independent) parameter; and note that if this parameter a is purely imaginary, Re [a]=0, Im [a] 6=0, then this dynamical system (75) is generally doubly periodic; or even—if a2/b2 is a real rational number— isochronous, namely then all its solutions are completely periodic with a period (an integer multiple of T =2π/ |a|) independent of the initial data: see Remark A.2-1 and, if need be, [7] [3]. Because of this remarkable fact, in the remaining part of this Section 5 we limit, for simplicity, consideration to the special case (72), by investigating its extension which obtains via the following linear reshuﬄe of the 2 dependent variables x1 (t) and x2 (t):

z1 = A11x1 + A12x2 , z2 = A21x1 + A22x2 , (76a) which is inverted to read as follows

x1 = (A22z1 − A12z2) /D, x2 = (−A21z1 + A11z2) /D ; (76b) here and hereafter D = A11A22 − A12A21 . (76c) It is easily seen that the new system then reads

2 2 z˙n = an1 (z1) + an2 (z2) + an3z1z2 , n =1, 2 , (77) with the 6 parameters anℓ, n = 1, 2, ℓ = 1, 2, 3 explicitly expressed in terms of the 4 arbitrary parameters Anm, n = 1, 2, m = 1, 2, and the 2 arbitrary parameters a2 and b2 (see (72c)) as follows:

−2 2 2 an1 = D (A22) (An1c11 + An2c21) + (A21) (An1c12 + An2c22)

−A22hA21 (An1c13 + An2c23)] , n =1, 2 , (78a)

−2 2 2 an2 = D (A12) (An1c11 + An2c21) + (A11) (An1c12 + An2c22)

−A11hA12 (An1c13 + An2c23)] , n =1, 2 , (78b)

−2 an3 = D [−2A12A22 (An1c11 + An2c21) − 2A21A11 (An1c12 + An2c22)

+ (A11A22 + A12A21) (An1c13 + An2c23)] , n =1, 2 . (78c)

27 Remark 5.3. The fact that the 6 parameters anℓ which characterize the system (77) can be (explicitly!) expressed, see (78), in terms of 6 a priori arbitrary parameters—the 4 parameters Anm, see (76), and the 2 parameters a2 and b2 (see (72))—might seem to imply that this system (77) can be reduced by algebraic operations to the algebraically solvable system (72)— hence that it is itself algebraically solvable—for any generic assignment of its 6 parameters anℓ, n = 1, 2, ℓ = 1, 2, 3. That this is not the case is however implied by the observation that the property of the system (72)—to feature in the right-hand sides of its 2 ODEs 2 polynomials themselves featuring a common zero (see Remark 5-1)—is then clearly also featured by the generalized system (77) (we like to thank Fran¸cois Leyvraz for this very useful observation). Hence only (at most) 5 of the 6 parameters anℓ (n =1, 2; ℓ =1, 2, 3) can be arbitrarily assigned, since these 6 parameters are constrained by the condition

2 (a11a22 − a21a12) + (a13a21 − a11a23) (a13a22 − a12a23)=0 (79) which is easily seen to correspond to the requirement that the right-hand sides of the 2 ODEs (77) (with n =1, 2) feature a common zero. Remark 5-4. Let us ﬁnally emphasize that the trick reported in Remark 5-1 is just as applicable to the more general system (77), implying—via the ansatz τ = exp(at); Zn (t) = exp(at) zn (τ) , n =1, 2 , (80a) analogous to (74)—the solvability of the system

2 2 Z˙ n = aZn + an1 (Z1) + an2 (Z2) + an3Z1Z2 , n =1, 2 , (80b) featuring the 7 parameters a and anℓ (n =1, 2; ℓ =1, 2, 3). The relevance of this dynamical system, (80b), in many applicative context is exempliﬁed by too many contributions to allow reporting a full bibliography; we record here just one such paper which lists 11 references and contains the remarkable assertion that the system (80b) ”is not solvable explicitly except in certain simple cases” [13].

6 Outlook

In this final Section 6 we tersely outline future developments of the findings reported in this paper. There is of course the possibility to treat cases with M > 4 (see Section 3). There is the possibility to iterate the procedure leading to the identification of new solvable systems (as described in this paper): see for this kind of development [14] and Chapter 6 of [7]. Another natural development is to treat analogous dynamical systems evolving in discrete rather than continuous time. For progress in this direction see [15].

28 Another extension is to treat systems characterized by second-order rather than first-order differential equations, including models characterized by New- tonian equations of motion (”accelerations equal forces”); and in the cases in which these equations of motion are derivable from a Hamiltonian, an additional interesting development is the treatment of the corresponding time-evolutions in the context of quantal rather than classical mechanics. And yet another extension is to Partial Differential Equations (PDEs) rather than ODEs. There is finally the vast universe of applications, including to cases in which the systems of evolution equations can be shown—via their solvability—to feature remarkable properties such as isochrony [2] [3] or asymptotic isochrony [4].

7 Acknowledgements

FP likes to thank the Physics Department of the University of Rome ”La Sapienza” for the hospitality from April to November 2018 (during her sabbat- ical), when the results reported in this paper were obtained. FC likes to thank Robert Conte and Fran¸cois Leyvraz for very useful discussions in the context of the Gathering of Scientists on ”Integrable systems and beyond” hosted by the Centro Internacional de Ciencias (CIC) in Cuernavaca, Mexico, from November 19th to December 14th, 2018.

8 Appendix A: Three useful classes of solvable systems of 2 nonlinear first-order ODEs for the 2 variables ym˜ (t) The findings reported in this Appendix A are not new; they are displayed here to facilitate the reader of the new findings reported in the body of this paper.

Notation A-1. In this Appendix A we indicate with the notation ym˜ 1 (t) and ym˜ 2 (t)—withm ˜ 1,2 =1, 2, 3, 4andm ˜ 1 6=m ˜ 2 (and for the signiﬁcance of the superimposed tilde see the last part of Section 2)—the 2 dependent variables which satisfy the ”solvable” system of 2 nonlinearly-coupled ODEs

y˙m˜ 1 = fm˜ 1 (ym˜ 1 ,ym˜ 2 ) , y˙m˜ 2 = fm˜ 2 (ym˜ 1 ,ym˜ 2 ) , (81) with the 2 functions fm˜ 1 (ym˜ 1 ,ym˜ 2 ) and fm˜ 2 (ym˜ 1 ,ym˜ 2 ) assigned—conveniently for our treatment in this paper (see Section 2 above)—so that the system (81) is ”solvable”. The precise meaning of the term ”solvable” shall be clear from the following. In this Appendix A αℓ, βℓ, γℓ are a priori arbitrary time-independent parameters, and L is an a priori arbitrary nonnegative integer.

29 The selection of the speciﬁc systems of 2 ODEs considered below is of course motivated by the treatment in the body of this paper, see in particular Sections 2 and 3.

8.1 Case A.1 L L ℓm˜ 2+1 ℓm˜ 1+1 y˙m˜ 1 = αℓ (ym˜ 1 ) , y˙m˜ 2 = βℓ (ym˜ 2 ) . (82) Xℓ=0 h i Xℓ=0 h i Each of these 2 ODEs can be integrated via one quadrature, that can be performed explicitly after some purely algebraic operations. Indeed, to integrate the first of these 2 ODEs one must first of all identify—via an algebraic operation— the Lm˜ 2 +1 zerosy ¯n (assumed below, for simplicity, to be all different among themselves) of the polynomial in its right-hand side,

L Lm˜ 2+1 ℓm˜ 2+1 αℓ (ym˜ 1 ) = αL (ym˜ 1 − y¯n) ; (83) n=1 Xℓ=0 h i Y next one must identify the Lm˜ 2 +1 ”residues” rn deﬁned by the ”partial fraction decomposition” formula

Lm˜ 2+1 Lm˜ 2+1 −1 −1 (ym˜ 1 − y¯n) = rn (ym˜ 1 − y¯n) (84) n=1 n=1 Y X h i —another algebraic operation, indeed one that can be performed explicitly; and ﬁnally one integrates the ODE getting the (generally implicit; but not always, see below) result

Lm˜ 2+1 rn y 1 (t) − y¯ t m˜ n = exp , (85) y 1 (0) − y¯ α n=1 m˜ n L Y which characterizes the solution ym˜ 1 (t) corresponding to the initial datum ym˜ 1 (0) . Of course an analogous procedure characterizes—for the second ODE (82)— the solution ym˜ 2 (t) corresponding to the initial datum ym˜ 2 (0) . For L = 1 the initial-value problem for these 2 ODEs can be solved explicitly, since the solution of the initial-value problem for the ODE

y˙ = Ay + ByM+1 (86a) is provided by the formula

−1 y (t)= y (0)exp(At) 1 + (B/A)[y (0)]M [1 − exp(MAt)] . (86b) n o

30 8.2 Case A.2

L ℓ y˙m˜ 1 = αℓ (ym˜ 1 ) , (87a) Xℓ=0 h i L L ℓ−1 ℓ−1+(m ˜ 2/m˜ 1) y˙m˜ 2 = ym˜ 2 βℓ (ym˜ 1 ) + γℓ (ym˜ 1 ) . (87b) Xℓ=1 h i Xℓ=0 h i The solution of the ﬁrst of these 2 ODEs, (87a), has been already discussed above, see Subsection Case A.1; hence in this Subsection Case A.2 we need to consider only the second ODE (87b). And since we are mainly interested in the case when the right-hand side of this ODE is polynomial, we shall limit our consideration below to the 3 subcases withm ˜ 1 = 1 and to the single case with m˜ 1 = 2, m˜ 2 = 4; except in the special case with all parameters γℓ vanishing, γℓ = 0, which we treat separately ﬁrstly (since it is an intermediate step to solve the more general case). In this special case the ODE (87b) reads

L ℓ−1 y˙m˜ 2 = ym˜ 2 βℓ (ym˜ 1 ) , (88a) Xℓ=1 h i with the function ym˜ 1 (t) to be considered known; hence the solution of the initial-value problem for this ODE reads

ym˜ 2 (t)= ym˜ 2 (0) F (t) (88b) with t L ′ ′ ℓ−1 F (t) = exp dt βℓ [ym˜ 1 (t )] . (88c) ( 0 " #) Z Xℓ=1 n o And it is then easily seen that the solution of the initial-value problem of the (more general) ODE (87b) reads

t L ′ ′ −1 ′ ℓ−1+(m ˜ 2/m˜ 1) ym˜ 2 (t)= F (t) ym˜ 2 (0) + dt [F (t )] γℓ [ym˜ 1 (t )] . " 0 # Z ℓ=0 n o X (89) More explicit solutions can be easily obtained in the following cases:

m˜ 1 =1 , m˜ 2 =2, 3, 4 orm ˜ 1 =2, m˜ 2 =4 ,M =m ˜ 2/m˜ 1 , (90a)

2 y˙m˜ 1 = α0 + α1ym˜ 1 + α2 (ym˜ 1 ) , (90b) −1+M M 1+M y˙m˜ 2 = ym˜ 2 (β0 + β1ym˜ 1 )+ γ0 (ym˜ 1 ) + γ1 (ym˜ 1 ) + γ2 (ym˜ 1 ) ; (90c)

ym˜ 1 (0) [1 + (∆/α1) tanh (∆t)] − 2 (α0/α1) tanh (∆t) ym˜ 1 (t) = , 1 − 2α2ym˜ 1 (0) + ∆ /α1 tanh (∆t) 2 2 ∆ = (α1) − 4α0α2 , (90d)

31 t L ′ ′ −1 ′ ℓ−1+(m ˜ 2/m˜ 1) ym˜ 2 (t)= f (t) ym˜ 2 (0) + dt [f (t )] γℓ [ym˜ 1 (t )] , " 0 # Z ℓ=0 n o X (91a) t L ′ ′ ℓ−1 f (t) = exp dt βℓ [ym˜ 1 (t )] . (91b) ( 0 " #) Z Xℓ=1 n o Remark A.2-1. In particular, it is easily seen that the system of ODEs (72) discussed in Section 5 (see Subsection 4.3 with a0 = a1 = a3 = b1 = b3 = 0) implies that the system of ODEs (87), which then reads

2 3 y˙1 = α2 (y1) , y˙2 = β2y2y1 + γ2 (y1) , (92a) with α2 = − (a2 +4b2) , β2 = −2a2 , γ2 =3b2 , (92b) features the following explicit solution of its initial-values problem:

y1 (0) y1 (t)= , (93a) 1 − α2y1 (0) t

2 −β2/α2 y2 (t) = y2 (0)+ (3/8) [y1 (0)] [1 − α2y1 (0) t] n 3 y (0) o2 − 1 . (93b) 8 1 − α y (0) t 2 1 The corresponding solution of the initial-values problem for the system of ODEs (72) is then obtained from this solution (93) via the relations y1 = −3x1 − x2, y2 =3x1 (x1 + x2) (see (12)), which of course imply

x2 (t)= −3x1 (t) − y1 (t) , (94a) with x1 (t) given in terms of y1 (t) and y2 (t) (see (93)) as a solution of the (explicitly solvable!) second-degree equation

2 6 (x1) +3y1x1 + y2 =0 . (94b)

8.3 Case A.3

−1+(m ˜ 2/m˜ 1) −1+2(m ˜ 2/m˜ 1) y˙m˜ 1 = α0 +α1ym˜ 2 , y˙m˜ 2 = β0 (ym˜ 1 ) +β1 (ym˜ 1 ) . (95a)

Since we are mainly interested in the case when the right-hand sides of both these ODEs are polynomial, we shall limit our consideration of this case to the 3 subcases withm ˜ 1 = 1 and to the single case withm ˜ 1 =2, m˜ 2 =4.

32 The most convenient technique to solve the system of 2 ODEs (95a) is by noticing to begin with that it implies for the variable ym˜ 1 (t) the second-order ODE −1+(m ˜ 2/m˜ 1) −1+2(m ˜ 2/m˜ 1) y¨m˜ 1 = α1 β0 (ym˜ 1 ) + β1 (ym˜ 1 ) , (95b) which is an autonomoush Newtonian equation of motion (”accelerationi equal force”) and is of course solvable by quadratures (as discussed in more detail in the following special cases).

And of course once ym˜ 1 (t) is known,y ˙m˜ 2 = [y ˙m˜ 1 (t) − α0] /α1 is as well known (see the ﬁrst of the 2 ODEs (95a)).

8.3.1 Case A.3.1

In this casem ˜ 1 =1, m˜ 2 =2, orm ˜ 1 =2, m˜ 2 =4, so that (95b) reads

3 w¨ = α1 β0w + β1w ; (96) here and below w (t) stands for ym˜ 1 (t) respectively ym˜ 2 (t) , in the 2 casesm ˜ 1 = 1, m˜ 2 =2, respectivelym ˜ 1 =2, m˜ 2 = 4. It is easily seen that the solution of the initial-value problem of this ODE reads as follows: w (t)= µ sn (λ t + ρ, k) , (97a) with the 4 parameters λ, µ, ρ and k determined by the following 4 formulas in terms of the 3 parameters α1, β0, β1 and the initial data w (0) = y1 (0), u (0) = y2 (0) respectively w (0) = y2 (0), u (0) = y4 (0) (in the 2 casesm ˜ 1 =1, m˜ 2 =2, respectivelym ˜ 1 =2, m˜ 2 = 4):

2 2 −α1β0β1 2 −2k β0 w (0) λ = 2 , µ = 2 , sn (ρ, k)= , 1+ k β1 (1 + k ) µ 2 2 2 2 c1k + c2 1+ k =0 , c1 = −2α1 (β0) , α (β )2 c = (α )2 [β + β u (0)]2 − α β β [w (0)]2 − 1 1 [w (0)]4 . (97b) 2 1 0 1 1 0 1 2 And of course correspondingly (see the ﬁrst of the 2 ODEs (95a))

−1 u (t) = (α1β1) [λµ cn (λt + ρ, k) dn (λt + ρ, k) − α1β0] , (98) where of course u (t) stands for y2 (t) respectively y4 (t) . Here sn(z, k) , cn(z, k) , dn(z, k) are the 3 standard Jacobian elliptic functions.

8.3.2 Case A.3.2

In this casem ˜ 1 =1, m˜ 2 =3, so that (95b) reads

2 5 y¨1 = α1 β0 (y1) + β1 (y1) , (99a) h i 33 implying

y1(t) dy = t , 3 3 C + (α1/3) y (2β0 + β1y ) y1Z(0) p2 3 3 C = [y1 (0)] − (α1/3) [y1 (0)] 2β0 + β1 [y1 (0)] . (99b) n o It is thus seen that in this case y1 (t) is a hyperelliptic function.

8.3.3 Case A.3.3

In this casem ˜ 1 =1, m˜ 2 =4, so that (95b) reads

3 7 y¨1 = α1 β0 (y1) + β1 (y1) , (100a) h i implying

y1(t) dy = t , 4 4 C + (α1/4) y (2β0 + β1y ) y1Z(0) p2 4 4 C = [y1 (0)] − (α1/4) [y1 (0)] 2β0 + β1 [y1 (0)] . (100b) n o It is thus again seen that in this case y1 (t) is a hyperelliptic function.

9 Appendix B

In this Appendix B we display—for the case M = 4 and N = 2—the expressions of the time-derivativesy ˙m˜ (t) of the coeﬃcients ym˜ (t) in terms of the time-derivativesy ˙m (t) of the coeﬃcients ym (t) and of the zeros xn (t), for the 2 cases with µ1 =3, µ2 = 1 respectively µ1 = µ2 = 2 (for this terminology, see Section 2). In case (i), with µ1 =3, µ2 =1, these relations read as follows:

2 2x1y˙2 +y ˙3 (x1) y˙2 − y˙4 x1y˙3 + 2y ˙4 y˙1 = − 3 , y˙1 = − 3 , y˙1 = 3 , (101a) 3 (x1) 2 (x1) (x1)

2 3 3 (x1) y˙1 +y ˙3 2 (x1) y˙1 − y˙4 2x1y˙3 + 3y ˙4 y˙2 = − , y˙2 = − 2 , y˙2 = − 2 , (101b) 2x1 (x1) (x1) 3 2 2 (x1) y˙1 − 2y ˙4 (x1) y˙2 + 3y ˙4 y˙3 = −3 (x1) y˙1 − 2x1y˙2 , y˙3 = , y˙3 = − , x1 2x1 (101c)

34 1 y˙ = 2 (x )3 y˙ + (x )2 y˙ , y˙ = (x )3 y˙ − x y˙ , 4 1 1 1 2 4 2 1 1 1 3 1 h i y˙ = − (x )2 y˙ +2x y˙ . (101d) 4 3 1 2 1 3 h i In case (ii), with µ1 = µ2 =2, these relations read as follows:

(x1 + x2)y ˙2 +y ˙3 x1x2y˙2 − y˙4 y˙1 = − 2 2 , y˙1 = − , (x1) + x1x2 + (x2) x1x2 (x1 + x2) x1x2y˙3 + (x1 + x2)y ˙4 y˙1 = 2 , (102a) (x1x2)

2 2 (x1) + x1x2 + (x2) y˙1 +y ˙3 x1x2 (x1 + x2)y ˙1 − y˙4 y˙2 = − , y˙2 = − , h x1 + x2 i x1x2 2 2 x1x2 (x1 + x2)y ˙3 + (x1) + x1x2 + (x2) y˙4 y˙ = − , (102b) 2 h 2 i (x1x2)

2 2 y˙3 = − (x1) + x1x2 + (x2) y˙1 − (x1 + x2)y ˙2 , h 2 i (x1x2) y˙1 − (x1 + x2)y ˙4 y˙3 = , x1x2 2 2 2 (x1x2) y˙2 + (x1) + x1x2 + (x2) y˙4 y˙3 = − , (102c) xh1x2 (x1 + x2) i

2 (x1x2) y˙1 − x1x2y˙3 y˙4 = x1x2 (x1 + x2)y ˙1 + x1x2y˙2 , y˙4 = , x1 + x2 2 (x1x2) y˙2 + x1x2 (x1 + x2)y ˙3 y˙4 = − 2 2 . (102d) (x1) + x1x2 + (x2)

References

[1] F. Calogero, ”Motion of Poles and Zeros of Special Solutions of Nonlinear and Linear Partial Differential Equations, and Related ”Solvable” Many Body Problems”, Nuovo Cimento 43B, 177-241 (1978). [2] F. Calogero, Isochronous systems, Oxford University Press, Oxford, U. K., (2008); paperback (2012). [3] D. Gómez-Ullate and M. Sommacal, ”Periods of the goldfish many-body problem”, J. Nonlinear Math. Phys. 12, Suppl. 1, 351-362 (2005).

35 [4] F. Calogero and D. Gómez-Ullate, “Asymptotically isochronous systems”, J. Nonlinear Math. Phys. 15, 410-426 (2008). [5] F. Calogero, Classical many-body problems amenable to exact treatments, Lecture Notes in Physics Monograph m66, Springer, Heidelberg, 2001 (749 pages). [6] F. Calogero, “New solvable variants of the goldfish many-body problem”, Studies Appl. Math. 137 (1), 123-139 (2016); DOI: 10.1111/sapm.12096. [7] F. Calogero, Zeros of Polynomials and Solvable Nonlinear Evolution Equa- tions, Cambridge University Press, Cambridge, U. K., 2018 (168 pages). [8] O. Bihun and F. Calogero, “Novel solvable many-body problems”, J. Nonlinear Math. Phys. 23, 190-212 (2016). DOI: 10.1080/14029251.2016.1161260. [9] M. Bruschi and F. Calogero, “A convenient expression of the time- (k) derivative zn (t), of arbitrary order k, of the zero zn(t) of a time-dependent polynomial pN (z; t) of arbitrary degree N in z, and solvable dynamical systems”, J. Nonlinear Math. Phys. 23, 474-485 (2016). [10] O. Bihun and F. Calogero, “Time-dependent polynomials with one double root, and related new solvable systems of nonlinear evolution equations”, Qual. Theory Dyn. Syst. (published online: 26 July 2018). doi.org/10.1007/s12346-018-0282-3; http://arxiv.org/abs/1806.07502. [11] O. Bihun, ”Time-dependent polynomials with one multiple root and new solvable dynamical systems”, arXiv:1808.00512v1 [math-ph] 1 Aug 2018. [12] F. Calogero and F. Payandeh, ”Solvable dynamical systems in the plane with polynomial interactions”, to be published as a chapter in a collective book to celebrate the 65th birthdate of Emma Previato (in press). [13] G. R. Nicklason, ”The general phase plane solution of the 2D homogeneous system with equal Malthusian terms: the quadratic case”, Canadian Appl. Math Quarterly, 13 (1), 89-106 (2005). [14] O. Bihun and F. Calogero, “Generations of monic polynomials such that the coefficients of each polynomial of the next generation coincide with the zeros of a polynomial of the current generation, and new solvable many-body problems”, Lett. Math. Phys. 106 (7), 1011-1031 (2016); DOI: 10.1007/s11005-016-0836-8. arXiv: 1510.05017 [math-ph]. [15] F. Calogero and F. Payandeh, ”Solvable systems featuring 2 dependent variables evolving in discrete-time via 2 nonlinearly-coupled first-order re- cursion relations with polynomial right-hand sides”, J. Nonlinear Math. Phys. (submitted to, 21.11.2018).